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### Problem 1(a): 
**Define linear combination. Write the relevant picture of the following matrix:**

Given matrix equation:
\[
\begin{pmatrix}
2 & 4 \\
-3 & 8
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
= \begin{pmatrix}
5 \\
2
\end{pmatrix}
\]

#### Solution:
A **linear combination** of vectors is an expression constructed from a set of terms by multiplying each term by a constant and adding the results. 

Given the equation:
\[
\begin{pmatrix}
2 & 4 \\
-3 & 8
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
= \begin{pmatrix}
5 \\
2
\end{pmatrix}
\]
This corresponds to the system of linear equations:
\[
2x_1 + 4x_2 = 5
\]
\[
-3x_1 + 8x_2 = 2
\]

This system represents a linear combination of the column vectors \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\) and \(\begin{pmatrix} 4 \\ 8 \end{pmatrix}\) scaled by \(x_1\) and \(x_2\) respectively, resulting in the vector \(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\).

So, the "picture" of this linear combination can be written as:
\[
x_1 \begin{pmatrix} 2 \\ -3 \end{pmatrix} + x_2 \begin{pmatrix} 4 \\ 8 \end{pmatrix} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}
\]

### Problem 1(b): 
**Compute the dot product. What is the dot product of the following two vectors?**

Given vectors \(V = (1, 2)\) and \(W = (w_1, w_2)\).

#### Solution:
The **dot product** of two vectors \(V = (v_1, v_2)\) and \(W = (w_1, w_2)\) is given by:
\[
V \cdot W = v_1 \cdot w_1 + v_2 \cdot w_2
\]

Substituting the given values:
\[
V \cdot W = 1 \cdot w_1 + 2 \cdot w_2 = w_1 + 2w_2
\]

### Final Answers:
1. **Linear Combination**:
   \[
   x_1 \begin{pmatrix} 2 \\ -3 \end{pmatrix} + x_2 \begin{pmatrix} 4 \\ 8 \end{pmatrix} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}
   \]
   
2. **Dot Product**:
   \[
   V \cdot W = w_1 + 2w_2
   \]

Would you like further details on these solutions or have any specific questions?  

Here are some related questions to deepen your understanding:

1. What is the geometric interpretation of the dot product?
2. How do you solve the system of linear equations derived from a matrix equation?
3. Can a linear combination of two vectors result in the zero vector? If so, under what conditions?
4. What are the properties of the dot product in terms of vector magnitude and angle?
5. How does the determinant of a matrix relate to the solvability of the corresponding system of linear equations?

**Tip:** When working with systems of equations, it can be helpful to use matrix operations, such as row reduction, to simplify the solution process.

Explore the concepts of linear combination and dot product in matrix algebra. Learn how to solve problems involving linear equations and vector operations with detailed examples and explanations. Suitable for advanced high school students and anyone interested in linear algebra.

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